For permutations sampled from the Brownian separable permuton, LIS(σ_n)/n^{α(p)} converges almost surely to a positive finite random variable, and α(p) is the explicit solution of a Gamma-function equation.
Baxter permutations and plane bipolar orientations
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abstract
We present a simple bijection between Baxter permutations of size $n$ and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations (number of ascents, right-to-left maxima, left-to-right minima...) into natural parameters of plane bipolar orientations (number of vertices, degree of the sink, degree of the source...), and has remarkable symmetry properties. By specializing it to Baxter permutations avoiding the pattern 2413, we obtain a bijection with non-separable planar maps. A further specialization yields a bijection between permutations avoiding 2413 and 3142 and series-parallel maps.
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The longest increasing subsequence of Brownian separable permutons
For permutations sampled from the Brownian separable permuton, LIS(σ_n)/n^{α(p)} converges almost surely to a positive finite random variable, and α(p) is the explicit solution of a Gamma-function equation.